Kernel Distance Functions and Wavelets
Key words: kernel distance function, kernel
radial basis function, distance function wavelets, meshfree, multivariate
scattered data, Green second identity, translation invariant, rotational
invariant, ridge function, general solution, fundamental solution, kernel
geodesic distance function, partial differential equation, multifractal,
fractional derivative.
Albeit its
publicity and popularity, the radial basis function (RBF), which uses the
Euclidean distance variable underlying the rotational invariant, is only one of
many distance functions. For other examples, the distance functions involve the
translation invariant and inner product of two vectors. The present study is
not limited within the RBF and concerns general kernel distance functions and
distance function wavelets, establishing on the fundamental solution and the
general solution of partial differential equations (PDE). "Kernel"
here is closely related with the kernel functions of integral equations. Hereafter,
I will use the distance function instead of the RBF.
I first got to
know the distance function through my work in the dual reciproicty boundary
element method, where the distance function constitutes the very basis of its
prowess. The approach was originally introduced in the early 1970s to
multivariate scattered data approximations and function interpolations. Now it
is broadly employed in the neural network and machine learning, multivariate
scattered data processing, and in recent years, the fast emerging applications
in numerical PDE. Unlike the recently developed various meshfree FEM, BEM or
collocation methods which use moving least squares, the distance function
approach is an inherent cheap meshfree technique
Notably, the
distance function method is very mathematically simple to implement for
irregular high dimensional problems. Despite excellent performances in some
numerical experiments, reported work has been mostly based on intuitions. Two
bottlenecks are time-consuming evaluation of large dense interpolation matrix
and constructing efficient and reliable distance functions. By using the
multipole, moment or decomposition techniques, Prof. Beaston et al. have
presented the rapid solution schemes (Nlog(N)) to greatly reduce
computing cost. It is claimed that a single desktop PC (Pentium III) could now
handle distance function interpolation system of 5 million knots. But
nevertheless this kind of fast solution techniques has drawbacks in that they
lack the flexibility (basis funciton dependent) and requrie heavy programming.
Another challenging issue is to establish mathematical foundations of distance
functions. Albeit great effort, some limited advances are achieved.
The above issues
may be tackled in the framework of kernel distance functions, building on the
firm grounds of integral equation theory (distribution theory), and in terms of
physics, the potential theory. It is worth stressing that all kernel distance
functions are natural wavelet basis functions. Consequently, the distance function
wavelets will lead to a fast, efficient handling of various multifractal,
multiscale, multivariate, scattered data and numerical PDE problems. In
particular, the distance function wavelets is a natural tool to link the
fractional derivative and multifractal. As the motto goes "the laws of
universe are written in the language of partial differential equation",
the distance function and wavelets are not an exception. Below list my major
works on the distance function, wavelets and their applications from this
perspective.
Good reasons for "distance function" instead of
"radial basis function"
Kernel distance function wavelets transforms and series
Kernel distance functions and radial basis functions
Boundary knot method (Examplary
Matlab codes and geometric configurations)
Boundary particle method
Modified Kansa method
[ 
RBF || DQ-type methods
|| Modeling || BEM
|| Inverse analysis || Wavelet || Patent ]
![[HOME]](distance.files/image001.gif){kind=small}